Quasi-isometry invariance of percolation dimensions

Let $G$ and $H$ be finitely generated groups, and write $\operatorname{pdim}(G)$, $\nu\dim(G)$, and ${}^{e}\operatorname{pdim}(G)$ for the three percolation-dimension notions introduced in the source. A quasi-isometry between finitely generated groups is a map preserving distances up to uniform multiplicative and additive distortions. Quasi-isometry invariance conjecture. The quantities

$$\operatorname{pdim}(G),\quad \nu\dim(G),\quad {}^{e}\operatorname{pdim}(G)$$

are invariant under quasi-isometries between finitely generated groups. The source explicitly describes this as the most interesting currently open problem about these notions; proving it would establish that all three dimensions are geometric-group-theoretic invariants.